Module 2. Analitic geometry. Introduction to Calculus
Individual Home Task
Variant 5
1. For the triangle with the vertices А(–7;3), В(2;–1), С(–1;–5) find :
a) length of the side АВ;
b) equation of the line АМ which parallel to the side ВС;
c) the equation of the hieght ВF;
d) the equation of the median AD;
e) the interior angle ;
f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;
g) area of the treangular АВС.
2.
There
are given the points М1(1;–2;1),
M2(2;3;4),
of the plane Р1:
x–2y+3z–4=0,
Р2:
2x+3y–4z+5=0
and лінії
L1:
;
L2:
.
а) Write down the equation of the straight line passing through the points М1 and M2;
b) Find the acute angle between the lines L1 and L2;
c) Through the point M2 draw the plane parallel to the plane Р2;
d) Find the acute angle between the planes Р1 and Р2;
e) Find the distance from the point М1 to the plane Р2;
f) Through the point M2 draw the line parallel to the line L2;
g) Find the acute angle between the line L2 and the plane Р2;
h) Through the point М1 draw the line perpendicular to the plane Р1;
i) Write down the equation of the plane passing through the point М1 and perpendicular to the planes Р1 and Р2;
j) Write down the canonic equation of the line which is the concurent of two planes Р1 and Р2.
3.
Calculate the limits:
а)
;
b)
;
c)
;
d)
.
4.
Calculate the limits:
а)
;
b)
.
5.
Investigate the function for continuity:
Module 2. Analitic geometry. Introduction to Calculus
Individual Home Task
Variant 6
1. For the triangle with the vertices А(–8;–2), В(2;10), С(4;4) find :
a) length of the side АВ;
b) equation of the line АМ which parallel to the side ВС;
c) the equation of the hieght ВF;
d) the equation of the median AD;
e) the interior angle ;
f) the coordinates of the points N and K, which divides the greater sige of the triangular into three equal parts;
g) area of the treangular АВС.
There are given the points М1(1;2;–6), M2(–3;3;4), of the plane Р1: x+y–z+4=0, Р2: 2x-y+4z+5=0 and лінії L1:
;
L2:
.
a) Write down the equation of the straight line passing through the points М1 and M2;
b) Find the acute angle between the lines L1 and L2;
c) Through the point M2 draw the plane parallel to the plane Р2;
d) Find the acute angle between the planes Р1 and Р2;
e) Find the distance from the point М1 to the plane Р2;
f) Through the point M2 draw the line parallel to the line L2;
g) Find the acute angle between the line L2 and the plane Р2;
h) Through the point М1 draw the line perpendicular to the plane Р1;
i) Write down the equation of the plane passing through the point М1 and perpendicular to the planes Р1 and Р2;
j) Write down the canonic equation of the line which is the concurent of two planes Р1 and Р2.
3.
Calculate the limits:
а)
;
b)
;
c)
;
d)
.
4.
Calculate the limits:
а)
;
b)
.
5.
Investigate the function for continuity:
.